A303762 - OEIS

%I #33 Jun 07 2018 22:10:11

%S 1,2,6,3,15,5,10,30,210,105,35,7,14,42,21,231,77,11,22,66,33,165,55,

%T 110,330,2310,1155,385,770,154,462,6006,3003,1001,143,13,26,78,39,195,

%U 65,130,390,2730,1365,455,91,182,546,273,4641,1547,221,17,34,102,51,255,85,170,510,3570,1785,595,119,238,714,357,3927,1309,187,374,1122,561,2805,935

%N a(0) = 1, and for n >= 1, a(n) is either the largest divisor of a(n-1) not already present in the sequence, or (if all divisors already used), a(n-1) * {the least prime p such that p does not divide a(n-1) and p*a(n-1) is not already present}.

%C Each a(n+1) is either a divisor or a multiple of a(n).

%C The construction is otherwise like that of A303760, except here we choose the largest divisor instead of the smallest one. In contrast to A303760, this sequence is NOT permutation of A005117: 70 = A019565(13) is the first missing squarefree number. See also comments in A303769, A303749 and A302775.

%C Index of greatest prime factor of a(n) is monotonic and increments at n = {0, 1, 2, 4, 8, 15, 31, 50, 102, 157, 317, 480, 964, 1451, 2907, 4366, 8738, 13113, 26233, 39356, ...} - _Michael De Vlieger_, May 22 2018

%H Antti Karttunen, <a href="/A303762/b303762.txt">Table of n, a(n) for n = 0..26232</a>

%F a(n) = A019565(A303769(n)). [Conjectured]

%e From _Michael De Vlieger_, May 23 2018: (Start)

%e Table below shows the initial 31 terms at right. First column is index n. Second shows "." if a(n) = largest divisor of a(n-1), or factor p. Third shows presence "1" or absence "." of prime k among prime divisors of a(n).

%e    n   p\d    MN(n)      a(n)

%e   ---------------------------

%e    0     .    .            1

%e    1     2    1            2

%e    2     3    11           6

%e    3     .    .1           3

%e    4     5    .11         15

%e    5     .    ..1          5

%e    6     2    1.1         10

%e    7     3    111         30

%e    8     7    1111       210

%e    9     .    .111       105

%e   10     .    ..11        35

%e   11     .    ...1         7

%e   12     2    1..1        14

%e   13     3    11.1        42

%e   14     .    .1.1        21

%e   15    11    .1.11      231

%e   16     .    ...11       77

%e   17     .    ....1       11

%e   18     2    1...1       22

%e   19     3    11..1       66

%e   20     .    .1..1       33

%e   21     5    .11.1      165

%e   22     .    ..1.1       55

%e   23     2    1.1.1      110

%e   24     3    111.1      330

%e   25     7    11111     2310

%e   26     .    .1111     1155

%e   27     .    ..111      385

%e   28     2    1.111      770

%e   29     .    1..11      154

%e   30     3    11.11      462

%e   31    13    11.111    6006

%e   ...  (End)

%t Nest[Append[#, Block[{d = Divisors@ #[[-1]], p = 2}, If[Complement[d, #] != {}, Complement[d, #][[-1]], While[Nand[Mod[#[[-1]], p] != 0, FreeQ[#, p #[[-1]] ] ], p = NextPrime@ p]; p #[[-1]] ] ] ] &, {1}, 75] (* _Michael De Vlieger_, May 22 2018 *)

%o (PARI)

%o default(parisizemax,2^31);

%o up_to = 2^14;

%o A053669(n) = forprime(p=2, , if (n % p, return(p))); \\ From A053669

%o v303762 = vector(up_to);

%o m_inverses = Map();

%o prev=1; for(n=1,up_to,fordiv(prev,d,if(!mapisdefined(m_inverses,(prev/d)),v303762[n] = (prev/d);mapput(m_inverses,(prev/d),n);break)); if(!v303762[n], apu = prev; while(mapisdefined(m_inverses,try = prev*A053669(apu)), apu *= A053669(apu)); v303762[n] = try; mapput(m_inverses,try,n)); prev = v303762[n]);

%o A303762(n) = v303762[n+1];

%Y Subset of A005117.

%Y Cf. also A019565, A302774, A302775, A303749, A303769.

%Y Cf. A303760, A303761 (variants).

%K nonn

%O 0,2

%A _Antti Karttunen_, May 03 2018